Recent studies on the super edge-magic deficiency of graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,\mid V\left(G\right)\mid +\mid E\left(G\right)\mid \}$ such that $f\left(u\right)+f\left(v\right)+f\left(uv\right)$ is a constant for each $uv\in E\left(G\right)$. Also, $G$ is said to be super edge-magic if $f\left(V\left(G\right)\right)=\{1,2,\dots ,\mid V\left(G\right)\mid \}$. Furthermore, the super edge-magic deficiency ${\mu }_s\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G\cup n{K}_1$ is super edge-magic or $+\mathrm{\infty }$ if there exists no such integer $n$. In this paper, we introduce the parameter $l\left(n\right)$ as the minimum size of a graph $G$ of order $n$ for which all graphs of order $n$ and size at least $l\left(n\right)$ have ${\mu }_s\left(G\right)=+\mathrm{\infty }$, and provide lower and upper bounds for $l\left(G\right)$. Imran, Baig, and Feunovucíková established that for integers $n$ with $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, ${\mu }_s\left(D_n\right)\le 3n\mathrm{/}2-1$, where $D_n$ is the cartesian product of the cycle $C_n$ of order $n$ and the complete graph $K_2$ of order $2$. We improve this bound by showing that ${\mu }_s\left(D_n\right)\le n+1$ when $n\ge 4$ is even. Enomoto, Lladó, Nakamigawa, and Ringel posed the conjecture that every nontrivial tree is super edge-magic. We propose a new approach to attak this conjecture. This approach may also help to resolve another labeling conjecture on trees by Graham and Sloane.